Copresheaf categories

Content created by Fredrik Bakke, Egbert Rijke, Daniel Gratzer and Elisabeth Stenholm.

Created on 2023-11-01.
Last modified on 2024-03-11.

module category-theory.copresheaf-categories where

open import category-theory.categories
open import category-theory.category-of-functors-from-small-to-large-categories
open import category-theory.functors-from-small-to-large-precategories
open import category-theory.large-categories
open import category-theory.large-precategories
open import category-theory.natural-transformations-functors-from-small-to-large-precategories
open import category-theory.precategories
open import category-theory.precategory-of-functors-from-small-to-large-precategories

open import foundation.category-of-sets
open import foundation.commuting-squares-of-maps
open import foundation.function-extensionality
open import foundation.function-types
open import foundation.homotopies
open import foundation.identity-types
open import foundation.sets
open import foundation.universe-levels

Idea

Given a precategory C, we can form its copresheaf category as the large category of functors from C, into the large category of sets

  C → Set.

To this large category, there is an associated small category of small copresheaves, taking values in small sets.

Definitions

The large category of copresheaves on a precategory

module _
  {l1 l2 : Level} (C : Precategory l1 l2)
  where

  copresheaf-large-precategory-Precategory :
    Large-Precategory (λ l → l1 ⊔ l2 ⊔ lsuc l) (λ l l' → l1 ⊔ l2 ⊔ l ⊔ l')
  copresheaf-large-precategory-Precategory =
    functor-large-precategory-Small-Large-Precategory C Set-Large-Precategory

  is-large-category-copresheaf-large-category-Precategory :
    is-large-category-Large-Precategory copresheaf-large-precategory-Precategory
  is-large-category-copresheaf-large-category-Precategory =
    is-large-category-functor-large-precategory-is-large-category-Small-Large-Precategory
      ( C)
      ( Set-Large-Precategory)
      ( is-large-category-Set-Large-Precategory)

  copresheaf-large-category-Precategory :
    Large-Category (λ l → l1 ⊔ l2 ⊔ lsuc l) (λ l l' → l1 ⊔ l2 ⊔ l ⊔ l')
  large-precategory-Large-Category copresheaf-large-category-Precategory =
    copresheaf-large-precategory-Precategory
  is-large-category-Large-Category copresheaf-large-category-Precategory =
    is-large-category-copresheaf-large-category-Precategory

  copresheaf-Precategory :
    (l : Level) → UU (l1 ⊔ l2 ⊔ lsuc l)
  copresheaf-Precategory =
    obj-Large-Precategory copresheaf-large-precategory-Precategory

  module _
    {l : Level} (P : copresheaf-Precategory l)
    where

    element-set-copresheaf-Precategory : obj-Precategory C → Set l
    element-set-copresheaf-Precategory =
      obj-functor-Small-Large-Precategory C Set-Large-Precategory P

    element-copresheaf-Precategory : obj-Precategory C → UU l
    element-copresheaf-Precategory X =
      type-Set (element-set-copresheaf-Precategory X)

    action-copresheaf-Precategory :
      {X Y : obj-Precategory C} →
      hom-Precategory C X Y →
      element-copresheaf-Precategory X → element-copresheaf-Precategory Y
    action-copresheaf-Precategory f =
      hom-functor-Small-Large-Precategory C Set-Large-Precategory P f

    preserves-id-action-copresheaf-Precategory :
      {X : obj-Precategory C} →
      action-copresheaf-Precategory {X} {X} (id-hom-Precategory C) ~ id
    preserves-id-action-copresheaf-Precategory =
      htpy-eq
        ( preserves-id-functor-Small-Large-Precategory C
          ( Set-Large-Precategory)
          ( P)
          ( _))

    preserves-comp-action-copresheaf-Precategory :
      {X Y Z : obj-Precategory C}
      (g : hom-Precategory C Y Z) (f : hom-Precategory C X Y) →
      action-copresheaf-Precategory (comp-hom-Precategory C g f) ~
      action-copresheaf-Precategory g ∘ action-copresheaf-Precategory f
    preserves-comp-action-copresheaf-Precategory g f =
      htpy-eq
        ( preserves-comp-functor-Small-Large-Precategory C
          ( Set-Large-Precategory)
          ( P)
          ( g)
          ( f))

  hom-set-copresheaf-Precategory :
    {l3 l4 : Level}
    (P : copresheaf-Precategory l3)
    (Q : copresheaf-Precategory l4) → Set (l1 ⊔ l2 ⊔ l3 ⊔ l4)
  hom-set-copresheaf-Precategory =
    hom-set-Large-Precategory copresheaf-large-precategory-Precategory

  hom-copresheaf-Precategory :
    {l3 l4 : Level}
    (X : copresheaf-Precategory l3) (Y : copresheaf-Precategory l4) →
    UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
  hom-copresheaf-Precategory =
    hom-Large-Precategory copresheaf-large-precategory-Precategory

  module _
    {l3 l4 : Level}
    (P : copresheaf-Precategory l3) (Q : copresheaf-Precategory l4)
    (h : hom-copresheaf-Precategory P Q)
    where

    map-hom-copresheaf-Precategory :
      (X : obj-Precategory C) →
      element-copresheaf-Precategory P X → element-copresheaf-Precategory Q X
    map-hom-copresheaf-Precategory =
      hom-natural-transformation-Small-Large-Precategory C
        ( Set-Large-Precategory)
        ( P)
        ( Q)
        ( h)

    naturality-hom-copresheaf-Precategory :
      {X Y : obj-Precategory C} (f : hom-Precategory C X Y) →
      coherence-square-maps
        ( action-copresheaf-Precategory P f)
        ( map-hom-copresheaf-Precategory X)
        ( map-hom-copresheaf-Precategory Y)
        ( action-copresheaf-Precategory Q f)
    naturality-hom-copresheaf-Precategory f =
      htpy-eq
        ( naturality-natural-transformation-Small-Large-Precategory C
          ( Set-Large-Precategory)
          ( P)
          ( Q)
          ( h)
          ( f))

  comp-hom-copresheaf-Precategory :
    {l3 l4 l5 : Level}
    (X : copresheaf-Precategory l3)
    (Y : copresheaf-Precategory l4)
    (Z : copresheaf-Precategory l5) →
    hom-copresheaf-Precategory Y Z →
    hom-copresheaf-Precategory X Y →
    hom-copresheaf-Precategory X Z
  comp-hom-copresheaf-Precategory X Y Z =
    comp-hom-Large-Precategory
      ( copresheaf-large-precategory-Precategory)
      { X = X}
      { Y}
      { Z}

  id-hom-copresheaf-Precategory :
    {l3 : Level} (X : copresheaf-Precategory l3) →
    hom-copresheaf-Precategory X X
  id-hom-copresheaf-Precategory X =
    id-hom-Large-Precategory copresheaf-large-precategory-Precategory {X = X}

  associative-comp-hom-copresheaf-Precategory :
    {l3 l4 l5 l6 : Level}
    (X : copresheaf-Precategory l3)
    (Y : copresheaf-Precategory l4)
    (Z : copresheaf-Precategory l5)
    (W : copresheaf-Precategory l6)
    (h : hom-copresheaf-Precategory Z W)
    (g : hom-copresheaf-Precategory Y Z)
    (f : hom-copresheaf-Precategory X Y) →
    comp-hom-copresheaf-Precategory X Y W
      ( comp-hom-copresheaf-Precategory Y Z W h g)
      ( f) ＝
    comp-hom-copresheaf-Precategory X Z W
      ( h)
      ( comp-hom-copresheaf-Precategory X Y Z g f)
  associative-comp-hom-copresheaf-Precategory X Y Z W =
    associative-comp-hom-Large-Precategory
      ( copresheaf-large-precategory-Precategory)
      { X = X}
      { Y}
      { Z}
      { W}

  left-unit-law-comp-hom-copresheaf-Precategory :
    {l3 l4 : Level}
    (X : copresheaf-Precategory l3)
    (Y : copresheaf-Precategory l4)
    (f : hom-copresheaf-Precategory X Y) →
    comp-hom-copresheaf-Precategory X Y Y
      ( id-hom-copresheaf-Precategory Y)
      ( f) ＝
    f
  left-unit-law-comp-hom-copresheaf-Precategory X Y =
    left-unit-law-comp-hom-Large-Precategory
      ( copresheaf-large-precategory-Precategory)
      { X = X}
      { Y}

  right-unit-law-comp-hom-copresheaf-Precategory :
    {l3 l4 : Level}
    (X : copresheaf-Precategory l3)
    (Y : copresheaf-Precategory l4)
    (f : hom-copresheaf-Precategory X Y) →
    comp-hom-copresheaf-Precategory X X Y
      ( f)
      ( id-hom-copresheaf-Precategory X) ＝
    f
  right-unit-law-comp-hom-copresheaf-Precategory X Y =
    right-unit-law-comp-hom-Large-Precategory
      ( copresheaf-large-precategory-Precategory)
      { X = X}
      { Y}

The category of small copresheaves on a precategory

module _
  {l1 l2 : Level} (C : Precategory l1 l2) (l : Level)
  where

  copresheaf-category-Precategory :
    Category (l1 ⊔ l2 ⊔ lsuc l) (l1 ⊔ l2 ⊔ l)
  copresheaf-category-Precategory =
    category-Large-Category (copresheaf-large-category-Precategory C) l

  copresheaf-precategory-Precategory :
    Precategory (l1 ⊔ l2 ⊔ lsuc l) (l1 ⊔ l2 ⊔ l)
  copresheaf-precategory-Precategory =
    precategory-Large-Precategory (copresheaf-large-precategory-Precategory C) l

See also

• The Yoneda lemma

External links

• Presheaf precategories at 1lab
• category of presheaves at $n$Lab
• copresheaf at $n$Lab

Recent changes

• 2024-03-11. Fredrik Bakke. Refactor category theory to use strictly involutive identity types (#1052).
• 2023-11-27. Elisabeth Stenholm, Daniel Gratzer and Egbert Rijke. Additions during work on material set theory in HoTT (#910).
• 2023-11-27. Fredrik Bakke. Refactor categories to carry a bidirectional witness of associativity (#945).
• 2023-11-19. Egbert Rijke. Codingstyle guideline that definitions should be specified (#920).
• 2023-11-09. Fredrik Bakke. Typeset nlab as $n$Lab (#911).